In the proposed RIS-aided cell-free network, all bss are synchronized, which is necessary to serve all users by coherent joint transmission [1]}. We follow the common assumption in the literature that all bss transmit the same symbols [2]}, [3]}, [4]}, [5]}. Let \({{\bf {s}}_{p}} \triangleq {\left[ {{s_{p,1}}, \cdots ,{s_{p,K}}} \right]^T} \in {{\mathbb {C}}^K}\) , where \({s}_{p,k}\)  denotes the transmitted symbol to the \(k\) -th user on the \(p\) -th subcarrier. We assume that the transmitted symbols have normalized power, i.e., \(\mathbb {E}\left\lbrace \mathbf {s}_{p}\mathbf {s}_{p}^{H}\right\rbrace =\mathbf {I}_{K}, \forall p \in \mathcal {P}\) . In the downlink, the frequency-domain symbol \({s}_{p,k}\)  is firstly precoded by the precoding vector \({\bf w}_{b,p,k}\in {{\mathbb {C}}^{M}}\)  at the \(b\) -th bs, so the precoded symbol \({{\bf {x}}_{b,p}}\)  at the \(b\) -th bs on the \(p\) -th subcarrier can be written as
\({{\bf {x}}_{b,p}} = \sum \limits _{k = 1}^K {{{{\bf {w}}_{b,p,k}}{s_{p,k}}}}.\) 
